Surface Area and Volume Class 9: notes and important questions

Surface Area and Volume: revision notes

1. Cuboid and cube

Cuboid \(l \times b \times h\): total surface area \(2(lb + bh + hl)\); lateral area (four walls) \(2(l + b)h\); volume \(lbh\). Cube of edge \(a\): \(6a^2\), lateral \(4a^2\), volume \(a^3\).

2. Right circular cylinder

Radius \(r\), height \(h\): curved surface area \(2\pi rh\); total surface area \(2\pi r(r + h)\); volume \(\pi r^2h\). (Unroll the curved surface: a rectangle \(2\pi r\) by \(h\).)

3. Right circular cone

Radius \(r\), height \(h\), slant height \(l = \sqrt{r^2 + h^2}\): curved surface area \(\pi rl\); total surface area \(\pi r(l + r)\); volume \(\dfrac13\pi r^2h\), one third of the cylinder with the same base and height.

4. Sphere and hemisphere

Sphere of radius \(r\): surface area \(4\pi r^2\); volume \(\dfrac43\pi r^3\). Hemisphere: curved surface \(2\pi r^2\), total surface (with the flat face) \(3\pi r^2\), volume \(\dfrac23\pi r^3\).

5. Units and scaling

• Area in square units, volume in cubic units; \(1\text{ m}^3 = 1000\) litres and \(1000\text{ cm}^3 = 1\) litre.
• If every length is multiplied by \(k\), areas are multiplied by \(k^2\) and volumes by \(k^3\).
• When a solid is melted and recast, the volume stays the same.

Worked example 1

Find the curved surface area of a cylinder with radius \(7\) cm and height \(10\) cm (take \(\pi = \dfrac{22}{7}\)).

\(2 \times \dfrac{22}{7} \times 7 \times 10 = 440\) cm².

Worked example 2

A cone has radius \(5\) cm and height \(12\) cm. Find its slant height and volume, in terms of \(\pi\).

\(l = 13\) cm; \(V = \dfrac13\pi \times 25 \times 12 = 100\pi\) cm³.

Worked example 3

The surface area of a sphere is \(616\) cm² (\(\pi = \dfrac{22}{7}\)). Find its radius.

\(4 \times \dfrac{22}{7}r^2 = 616 \Rightarrow r^2 = 49 \Rightarrow r = 7\) cm.

Common errors

• Using the height \(h\) instead of the slant height \(l\) in \(\pi rl\), or \(l\) instead of \(h\) in the cone's volume.
• Using the diameter as the radius.
• Forgetting the flat face in the total surface area of a hemisphere (\(3\pi r^2\), not \(2\pi r^2\)).
• Mixing m and cm, or cm³ and litres.

Exam tips

• Write the formula, then substitute; cancel the \(7\) in \(\dfrac{22}{7}\) before multiplying.
• Read whether the question wants curved, total or lateral surface area.